The tag has no wiki summary.

learn more… | top users | synonyms

12
votes
0answers
189 views

How large are the smallest-area projections of a high-dimensional convex body?

Let $B$ be a convex body in $\mathbb{R}^d$, equipped with its standard Euclidean form, and assume that $$\intop_B x \, dx = 0$$ $$\frac{1}{|B|_d} \intop_B x_i x_j \, dx = \delta_{ij},$$ a ...
7
votes
1answer
176 views

Uniform convergence of convex functions

It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform ...
6
votes
2answers
220 views

Continuous functions with convex level sets

Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set. To what extent such functions are studied? In particular: Is there ...
6
votes
2answers
172 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
5
votes
2answers
133 views

When does a cone contain its dual cone?

Let $V$ be a finite-dimensional vector space with an inner-product $(,)$ and let $C\subset V$ be a cone in $V$. Let $C^\vee$ denote the dual of $C$ with respect to $(,)$, i.e., the set of vector $v\in ...
5
votes
2answers
329 views

Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
5
votes
0answers
91 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
5
votes
0answers
141 views

Strong convexity of the trace of the square root of a matrix function

Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
4
votes
2answers
509 views

Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of ...
4
votes
3answers
372 views

When is a sequentially closed cone, closed?

The following question I also posed here, but still got no answer. Let $X$ be a locally convex, Hausdorff topological vector space and $C\subseteq X$ a convex cone, which is sequentially closed. What ...
4
votes
1answer
459 views

Quantitative Version of Jensen's Inequality?

Hi, I've been looking at situations where Jensen's inequality is almost tight, and found myself proving a lemma that I'm nearly certain exists somewhere in the literature. The specifics are as ...
4
votes
1answer
187 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
3
votes
2answers
147 views

Perimeter of a 'trapped' convex set

Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit ...
3
votes
3answers
235 views

infinite dimensional polyhedra

I have a reference request which I hope some reader here can help me with. I have encountered a set that has all the properties that one would expect from a polyhedral set (in the sense of finite ...
3
votes
1answer
76 views

Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...
3
votes
2answers
75 views

Is the prox-residual monotone?

$\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$ Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let ...
3
votes
1answer
65 views

generalized mean inequality extension

from generalized inequality, we now that for $p>q$, we have $M_p(\mathbf{x})\ge M_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ ...
3
votes
1answer
382 views

Possible to find a set of log-concave functions with log-concave sums?

While the set of log-convex functions is closed under addition, the set of log-concave functions is not. Yet if $f$ is log-concave, $\ln(k f) = \ln(k)+\ln(f)$, with $k \in \mathbb{R}^+$ constant, is ...
3
votes
1answer
208 views

Is there a non-compact Poulsen simplex?

A Choquet simplex is a closed, convex and metrizable subset of a locally convex Hausdorff topological vector space in which every point is a barycenter of a unique probability measure supported on the ...
3
votes
1answer
147 views

$\mathcal{H}$-polyhedron under a linear map

Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$. Moreover, let $M \colon \mathbb{R}^n \to ...
3
votes
1answer
158 views

Characterization of a set in $\mathbb{R}^d$

Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set. \begin{equation}\label{main12} C= \{x\in \mathbb{R}^d ~|~ ...
3
votes
2answers
253 views

The epigraph of a semi-convex function has positive reach

I've been trying to prove the following theorem for several hours with no result so far. Claim. Let $f:\mathbb{R} \to \mathbb{R}$ be a semi-convex function, i.e. there exists a constant $C > 0$ ...
3
votes
1answer
52 views

Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
3
votes
2answers
107 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
3
votes
0answers
139 views

Decreasing sequence of closed convex sets in a Banach space

Let $(C_n)$ be a decreasing sequence of closed convex subsets in a Banach space $(E,\Vert \cdot \Vert)$. The question I have is about the content of $C=\bigcap_{n=0}^\infty C_n$. If the $C_n$ are ...
3
votes
0answers
47 views

Minimax-like theorems involving union and intersection of regions in $\mathbb R^d$

From Minimax theorems, we roughly know that if $f(x,y)$ is convex on $X$ and concave $Y$ for both compact $X,Y$, then: $$ \max_{x\in X}\min_{y\in Y} f(x,y)=\min_{y\in Y}\max_{x\in X} f(x,y). $$ We can ...
3
votes
0answers
163 views

Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality $$ ...
2
votes
5answers
246 views

Distance between two sets

Let $A, B$ be two convex and closed subsets of $\mathbb{R}^n$. We would like to the minimum distance between these two sets. i.e., we want to find a solution for the following problem. $$ \min ...
2
votes
2answers
103 views

Projection onto rotated box

Does anyone know if there is an efficient way to find the projection of an arbitrary point $z$ onto a rotated box, i.e. onto the set $\Omega=\{x \mid a \leq Ux \leq b\}$ where $U$ is a unitary matrix? ...
2
votes
1answer
680 views

Projection exists => Uniformly convex?

Hello, I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x ...
2
votes
1answer
629 views

Integral inequality for convex function

Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true: \begin{equation} \frac{1}{b-a} \int_a^b ...
2
votes
2answers
277 views

A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am ...
2
votes
1answer
1k views

Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...
2
votes
1answer
475 views

existence of a minimum for a convex functional on a non-reflexive space

Let $X$ be a Banach space; $K\subset X$ nonempty, closed and convex; and $f:K\to \mathbb R$ lower semicontinuous, convex functional. Let also $f$ be coercive, i.e., $f(x)\to +\infty$ as $\|x\|\to ...
2
votes
1answer
123 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
2
votes
1answer
104 views

Known Results on Convexity of Numerical Range

Let $A_1,A_2,\dots,A_M$ be given $N\times N$ hermitian matrices. The numerical range is defined as the set \begin{align} \mathbb{S}=\{(u^HA_1u,\dots,u^HA_Mu)\in \mathbb{R}^M\mid u^Hu=1\} \end{align} ...
2
votes
1answer
441 views

Proving that a specific function is quasiconvex

Hello all, Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function ...
2
votes
0answers
50 views

A convex analysis theorem improvement

John's theorem states that to any full-dimensional symmetric convex set $K\subseteq R^n$ and any Ellipsoid $E\subseteq R^n$ that is centered at origin, there exists an invertible linear map $T$ so ...
2
votes
0answers
106 views

error estimate of linear interpolation in high dimension

Consider convex functions $f,g$ on $[0,1]^d$. Let $x_1,\cdots,x_n$ be $n\geq d+1$ fixed point in $[0,1]^d$ that is equally 'distributed' in the sense that $$c_1\leq ...
2
votes
0answers
325 views

On increasing the penalty term in convex optimization with regularization

Given the two strictly convex (unique solution) optimization problems as: $$Problem\:1:\min_{X} f(X)+\|X\|_{F}^2 \hspace{2cm}Problem \:2:\min_{X}f(X)+n\|X\|_2^2$$ where $X\in\mathbf{S}_{++}^{n}$ ...
2
votes
0answers
103 views

Lower convex envelope of a function involving entropy

Suppose two discrete random variables $X$ and $Y$ defined on finite sets $\mathcal{X}$ and $\mathcal{Y}$ are given and also suppose the conditional distribution $P_{Y|X}$ (i.e, channel) is fixed. We ...
2
votes
0answers
146 views

Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$

Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$. (Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$). Let $v ...
2
votes
0answers
133 views

Convex functions with non-singular hessian measure are continuously differentiable?

It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
2
votes
0answers
202 views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
2
votes
0answers
88 views

How to prove convexity for a complex integral including the variable in both limits and integrand?

During my research [on inventory management policies, i.e., something really applied ;-) ] I stumbled on integrals of the following type and I'm curious under which circumstances there are convex. ...
1
vote
2answers
186 views

A question involving Mazur's Lemma

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): "Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
1
vote
2answers
111 views

Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants \begin{align} w^{H}C_1w>0 \\\ w^{H}C_2w>0 \\\ ...~~~~~~~~~~ \\\ ...
1
vote
1answer
170 views

Are all points x of the boundary of a convex set C of a Hilbert space H projections onto C of a point different than x?

It appears that this is not true if H is of infinite dimension. My question is therefore the following: does anyone have a counter-example? Is there a caracterisation for the points of the boundary ...
1
vote
1answer
201 views

positive semigroups and convex operator

Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice V. (endowed with ordering $\leq$). $\phi:V\rightarrow V$ is a convex operator. I want to prove that ...
1
vote
1answer
214 views

How to examine the convexity of a complex function numerically?

I have a function which does not have a closed form . Large numerical effort should be done to evaluate the function for even a single point. How can I examine the convexity of my function over the ...